Use fixed-point iteration method to determine a solution accurate to within for on [1,2] use

The iteration we will consider is generated from the rearrangement:

then:

is our iteration.

Now we expand as a Taylor series about the root:

where is the remainder term, and as the serier is alternating from the second term onwards we know that this is bound by the absolute value of the first neglected term: (please justify this last inequality yourself, or something tighter if you like)

Hence as on we have

and so:

Hence in the iteration the error more than halves at each pass (in fact it does better than this I leave it to the reader to find tighter bounds here).

Hence the error after passes through the iteration is less than and we need to find such that:

this will guarantee that our error is less than the required limit, which looks like 6 iterations to me.

(in fact this is pessimistic since I have been rather generous in estimating the bounds)